Implicit improved vertex control for uncertain,time-varying linear discrete-time systems with state and control constraints

The problem of regulating an uncertain and/or time-varying linear discrete-time system with state and control constraints to the origin is addressed. It is shown that feasibility and a robustly asymptotically stable closed loop can be achieved using an interpolation technique. The design method can be seen as an alternative to optimization-based control schemes such as Robust Model Predictive Control. Especially for problems requiring complex calculations to find the optimal solution, the present method can provide a straightforward suboptimal solution. A simulation demonstrates the performance of this class of constrained controllers.     

Optimal control for linear systems with state equality constraints

This paper deals with the optimal control problem for linear systems with linear state equality constraints. For deterministic linear systems, first we find various existence conditions for constraining state feedback control and determine all constraining feedback gains, from which the optimal feedback gain is derived by reducing the dimension of the control input space. For systems with stochastic noises, it is shown that the same gain used for constraining the deterministic system also optimally constrains the expectation of states inside the constraint subspace and minimizes the expectation of the squared constraint error. We compare and discuss performance differences between unconstrained (using penalty method), projected, and constrained controllers for both deterministic and stochastic systems. Finally, numerical examples are used to demonstrate the performance difference of the three controllers.     

Relatively optimal control and its linear implementation

Motivated by the fact that determining a feedback solution for the optimal control problem under constraints is a hard task we introduce the concept of relative optimality, roughly optimality for a specific (nominal) plant initial condition. We consider a generic discrete-time finite-horizon constrained optimal control problem for linear systems, and we seek for a state feedback (possibly dynamic) controller. As a fundamental requirement, we do not admit preactions or controller-state initialization based on the plant initial state and we assume our controller to be time-invariant. In particular, we do not consider controllers simply achieved by the feedforward and tracking of the optimal trajectory. A relatively optimal control is a stabilizing controller such that, if initialized at its zero state, produces the optimal (constrained) trajectory for the nominal initial condition of the plant. We show that one of such controllers is linear, dead-beat, and its order is equal to the length of the horizon minus the plant order, thus, of complexity which is known a priori. Some additional features such as the assignment of the compensator poles to achieve strong stabilization are proposed. We show that, by means of the proposed approach, we can face several problems such as optimal point-to-point operations, optimal impulse response and optimal tracking.     

Explicit reference governor for linear systems

The explicit reference governor is a constrained control scheme that was originally introduced for generic nonlinear systems. This paper presents two explicit reference governor strategies that are specifically tailored for the constrained control of linear time-invariant systems subject to linear constraints. Both strategies are based on the idea of maintaining the system states within an invariant set which is entirely contained in the constraints. This invariant set can be constructed by exploiting either the Lyapunov inequality or modal decomposition. To improve the performance, we show that the two strategies can be combined by choosing at each time instant the least restrictive set. Numerical simulations illustrate that the proposed scheme achieves performances that are comparable to optimisation-based reference governors.     

Dynamic state feedback and its application to linear optimal control

We investigate the linear optimal control problems in the context of dynamic state feedback configuration. The dynamic state feedback is a dual structure of the dynamic observer. In conjunction with the well known arguments on linear matrix differential and Lyapunov equations, we elicit the fact that the quadratic performance index is always computable with this configuration. Based on this property, we suggest a nonlinear optimization programming method to get suboptimal or near optimal time-invariant dynamic state feedback controls. One can also evaluate the efficacy of pre-designed dynamic state feedback controllers utilizing this property. Two illustrative examples are given to show the effectiveness of the proposed approach.     

Optimal control of LTI systems over unreliable communication links

In this paper, optimal control of linear time-invariant (LTI) systems over unreliable communication links is studied. The motivation of the problem comes from growing applications that demand remote control of objects over Internet-type or wireless networks where links are prone to failure. Depending on the availability of acknowledgment (ACK) signals, two different types of networking protocols are considered. Under a TCP structure, existence of ACK signals is assumed, unlike the UDP structure where no ACK packets are present. The objective here is to mean-square (m.s.) stabilize the system while minimizing a quadratic performance criterion when the information flow between the controller and the plant is disrupted due to link failures, or packet losses. Sufficient conditions for the existence of stabilizing optimal controllers are derived.     

多通道约束下的网络控制系统的最优跟踪性能

本文针对双通道约束下的线性时不变网络控制系统的随机信号跟踪性能极限问题进行了研究.网络通信包含通信噪声和通信带宽两种信道因素.被控系统考虑是非最小相位和不稳定系统,并且系统包含多个不同的非最小相位零点和多个不同的不稳定极点.对上行通道和下行通道都存在通信带宽约束及高斯白噪声影响的情形,从频域角度,通过采用双自由度控制器和尤拉参数化方法,获得了此类网络控制系统的最优可达的跟踪性能.研究结果表明网络控制系统的跟踪性能极限完全由被控对象的结构特征(非最小相位零点、不稳定极点以及被控对象的系统增益),参考输入信号和网络特性(高斯白噪声的统计特征、通信信道带宽)所决定.最后,仿真结果检证了所得结果的正确性.     

Distributed optimal control of multiple systems

This article considers distributed optimal control of multiple linear systems. Distributed approximately optimal controllers are proposed for each system with the aid of communications between systems. The proposed controllers make the states of the closed-loop systems exponentially converge to the states of the closed-loop systems with the centralised optimal controllers if the communication digraph is strongly connected. If the communication digraph is switching and there are communication delays, the proposed controllers also make the states of the closed-loop systems exponentially converge to the states of the closed-loop systems with the centralised optimal controllers. Simulation results show effectiveness of the proposed controllers.     

Uniting bounded control and MPC for stabilization of constrained linear systems

In this work, a hybrid control scheme, uniting bounded control with model predictive control (MPC), is proposed for the stabilization of linear time-invariant systems with input constraints. The scheme is predicated upon the idea of switching between a model predictive controller, that minimizes a given performance objective subject to constraints, and a bounded controller, for which the region of constrained closed-loop stability is explicitly characterized. Switching laws, implemented by a logic-based supervisor that constantly monitors the plant, are derived to orchestrate the transition between the two controllers in a way that safeguards against any possible instability or infeasibility under MPC, reconciles the stability and optimality properties of both controllers, and guarantees asymptotic closed-loop stability for all initial conditions within the stability region of the bounded controller. The hybrid control scheme is shown to provide, irrespective of the chosen MPC formulation, a safety net for the practical implementation of MPC, for open-loop unstable plants, by providing a priori knowledge, through off-line computations, of a large set of initial conditions for which closed-loop stability is guaranteed. The implementation of the proposed approach is illustrated, through numerical simulations, for an exponentially unstable linear system.     

Optimal structured static state-feedback control design with limited model information for fully-actuated systems

We introduce the family of limited model information control design methods, which construct controllers by accessing the plant’s model in a constrained way, according to a given design graph. We investigate the closed-loop performance achievable by such control design methods for fully-actuated discrete-time linear time-invariant systems, under a separable quadratic cost. We restrict our study to control design methods which produce structured static state feedback controllers, where each subcontroller can at least access the state measurements of those subsystems that affect its corresponding subsystem. We compute the optimal control design strategy (in terms of the competitive ratio and domination metrics) when the control designer has access to the local model information and the global interconnection structure of the plant-to-be-controlled. Finally, we study the trade-off between the amount of model information exploited by a control design method and the best closed-loop performance (in terms of the competitive ratio) of controllers it can produce.     

An optimal control problem for a class of distributed parameter systems

The optimal control of a class of linear deterministic time-invariant multi-dimensional distributed systems is considered. The unconstrained optimal control problem is formulated as a quadratic minimisation in a real Hilbert space. A conjugate gradient minimisation technique is employed in its solution. The effect of constraints on the control variables is included by adding penalty terms to the performance criterion. This reduces the constrained optimisation to a series of unconstrained minimisations in Hilbert space. A two-dimensional heat conduction system is then considered as an illustrative example. Values for its Green's function are obtained using a numerical technique.     

Constrained linear quadratic gaussian control with process applications

Constrained linear quadratic Gaussian control is considered. Important practical design constraints including restrictions in control signal variations and in regulator structure are introduced quantitatively into the control problem formulation. Various topics in the resulting extension of the standard LQG design procedure are discussed, for instance optimality conditions, design of optimal low-order controllers and variance-constrained self-tuning control. Numerical algorithms for solving the constrained LQG control problems are given facilitating the application of the design procedure. Three industrial applications of linear quadratic Gaussian design are described. The examples are taken from the cement industry and from a process for the production of plastic film.     

Uniformly optimal control of linear time-invariant plants: Nonlinear time-varying controllers

It is shown that in the problems of uniformly (or H^∞−) optimal control of linear time-invariant plants, arbitrary nonlinear, time-varying controllers offer no advantage over linear, time-invariant controllers.     

Lower order generalized aggregated model and suboptimal control

A method for reducing the order of a linear time-invariant dynamic system is presented. It is shown that it is possible to retain the predominant eigenvalues (or any other set of eigenvalues) of the exact system in the lower order model that possesses the property that its state is an aggregation of the state variables of the original system. Also it is shown that the output of the reduced order model can be constrained to contain all the modes of the exact output and to be close to the actual output of the original system within a specified tolerance. The performance of the original system is investigated for an optimal output regulator problem, when it is controlled on the assumption that its behavior is governed by that of the lower order model. Relations are obtained for the performance degradation that results with the above suboptimal control policy. Numerical examples show that the suboptimal control can be used in practice to lessen the computational complexity required for the higher order optimal control. The stability of the suboptimal control is not guaranteed; however, it is reasonable to expect it to be asymptotically stable when the order of reduction is not excessively high, because the outputs of the exact and lower order models are tolerably close.     

Leader-following control of multiple nonholonomic systems over directed communication graphs

This paper considers the leader-following control problem of multiple nonlinear systems with directed communication topology and a leader. If the state of each system is measurable, distributed state feedback controllers are proposed using neighbours’ state information with the aid of Lyapunov techniques and properties of Laplacian matrix for time-invariant communication graph and time-varying communication graph. It is shown that the state of each system exponentially converges to the state of a leader. If the state of each system is not measurable, distributed observer-based output feedback control laws are proposed. As an application of the proposed results, formation control of wheeled mobile robots is studied. The simulation results show the effectiveness of the proposed results.     

Multirate constrained adaptive control

An adaptive servo control law with input constraints is derived for application to linear time-invariant systems with unknown parameters and two sampling rates: a slower one for the output, and a faster one for the input. The error between the actual and a reference performance index is shown to be bounded by the product of a finite gain and the parameter estimation error in the limit sense. Sufficient conditions for parameter convergence are proven under which the performance of the multirate, adaptive constrained control system asymptotically approaches that of the analogous single (fast) rate, constrained control system with known, constant parameters. The advantages of the multirate, adaptive constrained control algorithm are demonstrated by numerical simulations.     

Simultaneous pole assignment in linear periodic systems byconstrained structure feedback

The authors present methods for pole assignment by feedback with constrained structure in linear periodic systems using generalized sampled-data hold functions (GSHF). Constrained structure is taken to mean that the input of each channel is restricted to depend only on the measurements of some specific output channels. The basic idea of GSHF control is to use sample and hold, but to consider the hold function as a design parameter. Four strategies are proposed for closing a control loop using GSHF. The key to the results obtained is that, when GSHF control with constrained structure is applied to a periodic system, a discrete-time time-invariant, decentralized system is obtained for which control design methods are available     

Jump linear quadratic Gaussian control in continuous time

The optimal quadratic control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes is addressed. The systems are also subject to Gaussian input and measurement noise. The optimal solution for the jump linear-quadratic-Gaussian (JLQC) problem is given. This solution is based on a separation theorem. The optimal state estimator is sample-path dependent. If the plant parameters are constant in each value of the underlying jumping process, then the controller portion of the compensator converges to a time-invariant control law. However, the filter portion of the optimal infinite time horizon JLQC compensator is not time invariant. Thus, a suboptimal filter which does converge to a steady-state solution (under certain conditions) is derived, and a time-invariant compensator is obtained     

Output constrained IMC controllers in control systems of electromechanical actuators

Electromechanical actuators are widely used in many industrial applications. There are usually some constraints existing in a designed system. This paper proposes a simple method to design constrained controllers for electromechanical actuators. The controllers merge the ideas exploited in internal model control and model predictive control. They are designed using the standard control system structure with unity negative feedback. The structure of the controllers is relatively simple as well as the design process. The output constraint handling mechanism is based on prediction of the control plant behavior many time steps ahead. The mechanism increases control performance and safety of the control plant. The benefits offered by the proposed controllers have been demonstrated in real-life experiments carried out in control systems of two electromechanical actuators: a DC motor and an electrohydraulic actuator.     

Time-varying feedback laws for decentralized control

Decentralized control schemes are considered for time-invariant, finite dimensional, linear systems with know state equations. It is assumed that the systems are reachable and observable at a fictitious centralized control station, and that there is strong connectivity between the decentralized control stations via the system where necessary. It is shown that whether or not there are decentralized fixed modes in the open-loop system, periodically varying feedback gains at all but one of the control stations permit the remaining control station to observe and control the system given knowledge of the control laws implemented at the other control stations. Certain time-invariant systems which cannot be stabilized by decentralized time-invariant controllers, namely those with unstable decentralized fixed modes, can thus be stabilized by decentralized time-varying controllers.